On the Tree Conjecture for the Network Creation Game

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC\u2703] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees. We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees

Tree Nash Equilibria in the Network Creation Game

In the network creation game with n vertices, every vertex (a player) buys a set of adjacent edges, each at a fixed amount {\alpha} > 0. It has been conjectured that for {\alpha} >= n, every Nash equilibrium is a tree, and has been confirmed for every {\alpha} >= 273n. We improve upon this bound and show that this is true for every {\alpha} >= 65n. To show this, we provide new and improved results on the local structure of Nash equilibria. Technically, we show that if there is a cycle in a Nash equilibrium, then {\alpha} < 65n. Proving this, we only consider relatively simple strategy changes of the players involved in the cycle. We further show that this simple approach cannot be used to show the desired upper bound {\alpha} < n (for which a cycle may exist), but conjecture that a slightly worse bound {\alpha} < 1.3n can be achieved with this approach. Towards this conjecture, we show that if a Nash equilibrium has a cycle of length at most 10, then indeed {\alpha} < 1.3n. We further provide experimental evidence suggesting that when the girth of a Nash equilibrium is increasing, the upper bound on {\alpha} obtained by the simple strategy changes is not increasing. To the end, we investigate the approach for a coalitional variant of Nash equilibrium, where coalitions of two players cannot collectively improve, and show that if {\alpha} >= 41n, then every such Nash equilibrium is a tree

On the Structure of Equilibria in Basic Network Formation

We study network connection games where the nodes of a network perform edge swaps in order to improve their communication costs. For the model proposed by Alon et al. (2010), in which the selfish cost of a node is the sum of all shortest path distances to the other nodes, we use the probabilistic method to provide a new, structural characterization of equilibrium graphs. We show how to use this characterization in order to prove upper bounds on the diameter of equilibrium graphs in terms of the size of the largest $k$-vicinity (defined as the the set of vertices within distance $k$ from a vertex), for any $k \geq 1$ and in terms of the number of edges, thus settling positively a conjecture of Alon et al. in the cases of graphs of large $k$-vicinity size (including graphs of large maximum degree) and of graphs which are dense enough. Next, we present a new swap-based network creation game, in which selfish costs depend on the immediate neighborhood of each node; in particular, the profit of a node is defined as the sum of the degrees of its neighbors. We prove that, in contrast to the previous model, this network creation game admits an exact potential, and also that any equilibrium graph contains an induced star. The existence of the potential function is exploited in order to show that an equilibrium can be reached in expected polynomial time even in the case where nodes can only acquire limited knowledge concerning non-neighboring nodes.Comment: 11 pages, 4 figure

The Price of Anarchy for Network Formation in an Adversary Model

We study network formation with n players and link cost \alpha > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player v incorporates the expected number of players to which v will become disconnected. We show existence of equilibria and a price of stability of 1+o(1) under moderate assumptions on the adversary and n \geq 9. As the main result, we prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of player pairs to be separated. For unilateral link formation we show a bound of O(1) on the price of anarchy for both adversaries, the constant being bounded by 10+o(1) and 8+o(1), respectively. For bilateral link formation we show O(1+\sqrt{n/\alpha}) for one adversary (if \alpha > 1/2), and \Theta(n) for the other (if \alpha > 2 considered constant and n \geq 9). The latter is the worst that can happen for any adversary in this model (if \alpha = \Omega(1)). This points out substantial differences between unilateral and bilateral link formation